A144215 Triangle read by rows: T(n,k) is the number of forests on n unlabeled nodes with all nodes of degree <= k (n>=1, 0 <= k <= n-1).

1, 1, 2, 1, 2, 3, 1, 3, 5, 6, 1, 3, 7, 9, 10, 1, 4, 11, 17, 19, 20, 1, 4, 15, 28, 34, 36, 37, 1, 5, 22, 52, 67, 73, 75, 76, 1, 5, 30, 90, 129, 144, 150, 152, 153, 1, 6, 42, 170, 264, 305, 320, 326, 328, 329, 1, 6, 56, 310, 542, 645, 686, 701, 707, 709, 710

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Rebecca Neville, Graphs whose vertices are forests with bounded degree, Graph Theory Notes of New York, LIV (2008), 12-21. [Wayback Machine link]

Formula

Column k is Euler transform of column k of A144528. - Andrew Howroyd, Dec 18 2020

Example

Triangle begins:
  1
  1 2
  1 2  3
  1 3  5  6
  1 3  7  9 10
  1 4 11 17 19 20
  1 4 15 28 34 36 37
  ...
From Andrew Howroyd, Dec 18 2020: (Start)
Formatted as an array to show the full columns:
========================================================
n\k  | 0 1  2   3    4    5    6    7    8    9   10
-----+--------------------------------------------------
   1 | 1 1  1   1    1    1    1    1    1    1    1 ...
   2 | 1 2  2   2    2    2    2    2    2    2    2 ...
   3 | 1 2  3   3    3    3    3    3    3    3    3 ...
   4 | 1 3  5   6    6    6    6    6    6    6    6 ...
   5 | 1 3  7   9   10   10   10   10   10   10   10 ...
   6 | 1 4 11  17   19   20   20   20   20   20   20 ...
   7 | 1 4 15  28   34   36   37   37   37   37   37 ...
   8 | 1 5 22  52   67   73   75   76   76   76   76 ...
   9 | 1 5 30  90  129  144  150  152  153  153  153 ...
  10 | 1 6 42 170  264  305  320  326  328  329  329 ...
  11 | 1 6 56 310  542  645  686  701  707  709  710 ...
  12 | 1 7 77 600 1161 1431 1536 1577 1592 1598 1600 ...
(End)

Prog

(PARI) \\ Here V(n, k) gives column k of A144528.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
MSet(p, k)={my(n=serprec(p, x)-1); if(min(k, n)<1, 1 + O(x*x^n), polcoef(exp( sum(i=1, min(k, n), (y^i + O(y*y^k))*subst(p + O(x*x^(n\i)), x, x^i)/i ))/(1-y + O(y*y^k)), k, y))}
V(n, k)={my(g=1+O(x)); for(n=2, n, g=x*MSet(g, k-1)); Vec(1 + x*MSet(g, k) + (subst(g, x, x^2) - g^2)/2)}
M(n, m=n)={Mat(vector(m, k, EulerT(V(n, k-1)[2..1+n])~))}
{ my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } \\ Andrew Howroyd, Dec 18 2020

Crossrefs

The final diagonal gives A005195.

Column k=2 is A000041.

Cf. A144528, A144529, A339788.

Sequence in context: A104468 A293003 A110062 * A254539 A283827 A122087

Adjacent sequences: A144212 A144213 A144214 * A144216 A144217 A144218

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 20 2008

Extensions

Terms a(29) and beyond from Andrew Howroyd, Dec 18 2020

Status

approved